Let L be a positive line bundle over a projective complex manifold X, Lp its tensor power of order p, H 0 (X, Lp ) the space of holomorphic sections of Lp , and Np the complex dimension of H 0 (X, Lp ). The determinant of a basis of H 0 (X, Lp ), together with some given probability measure on a weighted compact set in X, induces naturally a β-ensemble, i.e., a random Np -point process on the compact set. Physically, depending on X and the value of β, this general setting corresponds to a gas of free or interacting fermions on X and may admit an interpretation in terms of some random matrix models. The empirical measures, associated with such β-ensembles, converge almost surely to an equilibrium measure when p goes to infinity. We establish a large deviation theorem (LDT) with an effective speed of convergence for these empirical measures. Our study covers a large class of β-ensembles on a compact subset of the unit sphere Sn ⊂ Rn+1 or of the Euclidean space Rn .